An improved algorithm for finding a length-constrained maximum-density subtree in a tree

Given a tree T with weight and length on each edge, as well as a lower bound L and an upper bound U , the so-called length-constrained maximum-density subtree problem is to find a maximum-density subtree in T such that the length of this subtree is between L and U . In this study, we present an algorithm that runs in O(nU log n) time for the case when the edge lengths are positive integers, where n is the number of nodes in T , which is an improvement over the previous algorithms when U = Ω(log n). In addition, we show that the time complexity of our algorithm can be reduced to O(nL log n L ), when the edge lengths being considered are uniform.